15* XHI INTEGRATION all x e E, we have f \ f > f* dAi(x)/ = J / J If in particular E is a Hilbert space, then since there is a semilinear iso- trv ZH~»/(X) of E onto its dual E' (12.15), we can define the notion of a scalarly integrable mapping x\-+f(x) of X into E: this means that, for each z e E the complex-valued function x\-+ (f(x) |z) is integrable. If the function JCH-HI f (jc)|| is integrable, then by (13.10.4) there exists a unique element of E, denoted by f f(X) ^(x) and called the integral (or weak integral) of f, with the property that /f \ f (U.iU.OJ 1 I V. y A*V. / */ ' V for all z e E. PROBLEMS 1. If a mapping JCH-*f« of X into R'is scalarly ft-integrable, then the element of R1 is called the integral of the mapping and is denoted by I ix dp(x). If, for all x e X, fx belongs to a weakly closed convex set A in R1 and if X is integrable and p,(X) = 1, then the integral \txdp,(x) also belongs to A. (Use Problem 13 of Section 12.15.) 2. Let E be a separable real Frechet space and E' its dual (12.15). It follows from the Hafm-Banach theorem (Section 12.15, Problem 4) that the linear mapping CE which maps each z e E to the linear form z' H-> <z, z') on Ex, is an infective mapping of E into R1' and is continuous with respect to the product topology on RE/. If K is any compact subset of E, the restriction CE | K is a homeomorphism of K onto the subspace cE(K) of RE' (1Z3.6). (a) Suppose that K is a compact convex subset of E. A mapping f of X into K is said to be scalarly ^"integrable if, for each z' e E'} the real-valued function *h-» <f(*), zx> is /i-integrable. Using the remarks above and Problem 1, show that there then exists a unique vector z e K such that <z, z'> = J<f(x), z'> dp&) for all z' e E'. This vector z is denoted by Jf dp, or f f(x) dpfr), and is called the integral of f. In particular, if X = K and f is the identity mapping 1K, for any positive measure fi on K with total mass equal to 1, the element b^ = f z dp, of K is called the bary center of ft. (b) Let H be a compact subset of E. Show that the closed convex hull K of H iscoarsest of the partitions which are finer